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Each section of the SAT test is supposed to be distributed normally with a mean of 500 and a standard deviation of 100. Suppose 5 students in a class took the SAT math test. They received the following scores: 400, 450, 575, 600, and 625.
(a) Assuming a normal distribution, write out the likelihood function for estimating the mean μ and standard deviation σ of the average class score.
(b) Calculate the MLE estimate for μ and the standard deviation (σ) for this class.
(c) What is the value of the log-likelihood function at this estimate?
(d) What is the least-squares estimate for μ and σ in your data? Are there any differences? Why or why not?
(e) Using a likelihood-ratio test, test whether the MLE estimates from the class are significantly different from the national mean of 500 and national standard deviation of 100.
(Hint: Start by plugging those values into your log-likelihood function. Remember you are testing two restrictions).
Mean Absolute Deviation To avoid the problem of positive and negative deviations canceling out each other, we can use the Mean Absolute Deviation which is given by
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